Isoresidual fibrations and the birational geometry of moduli of pointed stable curves

TL;DR

The paper demonstrates that the pseudo-effective cone of divisors on the moduli space of stable n-pointed curves is non-polyhedral for n≥8, showing these spaces are not Mori Dream Spaces.

Contribution

It constructs an explicit extremal non-polyhedral ray of the dual cone of moving curves using maps on meromorphic strata of differentials, revealing new geometric properties.

Findings

Pseudo-effective cone of  is non-polyhedral for n

Moduli spaces  are not Mori Dream Spaces

Constructs extremal non-polyhedral rays via maps on meromorphic differentials

Abstract

We show that the pseudo-effective cone of divisors of $\overline{M}_{0,n}$ is not polyhedral for $n\geq 8$ by constructing an extremal non-polyhedral ray of the dual cone of moving curves via maps on meromorphic strata of differentials returning the residues at the poles of the parameterised differentials. An immediate corollary is that these spaces are not Mori Dream Spaces.